The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.
-8 < (3h + 1)/2 < 8
First, multiply each segment of the system of inequalities by color(red)(2) to eliminate the fraction while keeping the system balanced:
color(red)(2) xx -8 < color(red)(2) xx (3h + 1)/2 < color(red)(2) xx 8
-16 < cancel(color(red)(2)) xx (3h + 1)/color(red)(cancel(color(black)(2))) < 16
-16 < 3h + 1 < 16
Next, subtract color(red)(1) from each segment to isolate the h term while keeping the system balanced:
-16 - color(red)(1) < 3h + 1 - color(red)(1) < 16 - color(red)(1)
-17 < 3h + 0 < 15
-17 < 3h < 15
Now, divide each segment by color(red)(3) to solve for h while keeping the system balanced:
-17/color(red)(3) < (3h)/color(red)(3) < 15/color(red)(3)
-17/3 < (color(red)(cancel(color(black)(3)))h)/cancel(color(red)(3)) < 5
-17/3 < h < 5
Or
h > -17/3 and h < 5
Or, in interval notation:
(-17/3, 5)